#10106234

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31)(33,34,38,41)(35,39,45,48)(36,37,46,40)(42,47,43,44);(2,7)(4,9)(8,13)(12,14)(17,19,23,20)(18,24,22,25)(21,27,29,28)(26,31,30,32)(33,35,38,45)(34,39,41,48)(36,43,46,42)(37,44,40,47);(2,7)(3,10)(8,13)(11,16)(33,36,38,46)(34,40,41,37)(35,42,45,43)(39,44,48,47);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,20,23,19)(18,25,22,24)(21,27,29,28)(26,31,30,32)(33,36)(34,37)(35,43)(38,46)(39,44)(40,41)(42,45)(47,48);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,35)(34,39)(36,42)(37,47)(38,45)(40,44)(41,48)(43,46);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)(33,37)(34,36)(35,44)(38,40)(39,43)(41,46)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10013430);